Most Accurate Estimate of Surface Area of a Bottle?

What is the most accurate way to estimate the surface area of a cylindrical bottle that decreases in diameter from its wides point (the body) to the narrowest point ( the neck and cap)?

It seems like we did similar problems in calculus, however: 1. I don't remember any of it, and 2. I think you were always provided with the mathmatical expression of curve of the bottle so that you could calculate it by rotating it around the center axis or something.

I just have the actual physical bottles in front of me and want to know what the outer surface areas are.

My best guess so far is to just treat them as cylinders and not take into account the curved surface where it transitions from the body to the neck, but I would like to be as accurate as possible.

Is there some correlation between displacement of water and surface area I could calculate by dunking them?

There is no fixed relationship between the volume and the surface area. This will depend on the actual shape of the bottle. You'll have to resort to some practical approach:

1. Find some scotch tape that comes in a tiny width. Carefully wrap the bottle in tape (avoiding gaps and overlaps) and keep track of the length of tape used.

2. If you've got a mass balance with a good sensitivity, you can do a dip in a viscous fluid (like honey) and measure the mass of fluid on the bottle. Then using the mass stuck to a calibrated known area of identical glass you can find the area of the bottle.

3. Variation of above technique : cover bottle in sticky goop and roll it in a pan of tiny ball bearings (or equivalent). The balls will make a monolayer thick coating.

It is not clear how you can have a "cylinder" that decreases in diameter. Cylinders have a constant diameter by definition. Perhaps you have a cone? In that case you can look up formulas for the surface area of a cone, Google "frustum of a cone"